The Impact of Bootstrap Methods on Time Series Analysis

Let X1, . . . ,Xn be an observed stretch from a strictly stationary time series {Xt, t ∈ Z}; the assumption of stationarity implies that the joint probability law of (Xt,Xt+1, . . . ,Xt+k) does not depend on t for any k ≥ 0. Assume also that the time series is weakly dependent, i.e., the collection of random variables {Xt, t ≤ 0} is approximately independent of {Xt, t ≥ k} when k is large enough. An example of a weak dependence structure is given by mdependence under which {Xt, t ≤ 0} is (exactly) independent of {Xt, t ≥ k} whenever k > m; independence is just the special case of 0-dependence. Due to the dependence between the observations, even the most basic methods involved in applied statistical work suddenly become challenging; an elementary such example has to do with estimating the unknown mean μ = EXt of the time series. The sample mean Xn = n−1 ∑n t=1 Xt is the obvious estimator; however—and here immediately the difficulties crop up— it is not the most efficient. To see why, consider the regression Xt = μ+ t, that also serves as a definition for the t process; it is apparent that Xn is the Ordinary Least Squares Estimator of μ in this model. However, because of the dependence in the errors t, the Best Linear Unbiased Estimator (BLUE) of μ is instead obtained by a Generalized Least Squares argument; consequently, μ BLUE = (1′Γ−1 n 1)−11′Γ−1 n X, where X = (X1, . . . ,Xn)′, 1 = (1, . . . , 1)′, and Γn is the (unknown) covariance matrix of the vector X with i, j element given by γ(i− j) = Cov(Xi,Xj). ∗ Professor of Mathematics and Adjunct Professor of Economics, University of California, San Diego, La Jolla, CA 92093-0112, USA; e-mail: dpolitis@ucsd.edu. Research partially supported by NSF grant DMS-01-04059. Many thanks are due to Prof. E. Paparoditis (Cyprus) and J. Romano (Stanford) for constructive comments on this manuscript.